The poset \(D(m)\)  has a special structure. It is multichainable (Lemma REF ). A poset \(P\)  is multichainable if there
is an enumeration of the elements \((v_{xy})_{(x,y)\in L\times V}\)  of the vertices of \(P\)  where \(V\)  is finite, \(L\)  is totally ordered
such that for every local isomorphism \(f\)  of \(L\) , the map \((f,{Id})\) , where \({Id}\)  is the identity map on \(V\) , is a local isomorphism of
\(P\) . This notion was introduced in 1978 [1]} (and published in [2]} (subsection IV.3.3.2, p.341),
and subsequent papers, e.g., [3]}).
It allows us to point out hereditary classes which are hereditary w.q.o., resp., hereditary b.q.o. A class \(\mathfrak {C}\)  has this property if the class \(\mathfrak {C}\cdot Q\)  of structures belonging to \(\mathfrak {C}\)  and labelled by a w.q.o., resp., a b.q.o. \(Q\) , remains w.q.o., resp., b.q.o. The age \(Age(R)\)  of a relational structure is the collection of the finite induced substructures of \(R\)  considered up to isomorphy. If \(R\)  is a multichainable structure, in particular a multichainable poset, then, with the help of Laver's theorem on countable chains, we get that the collection of countable structures whose age is included in \(Age(R)\)  is hereditary b.q.o.(Proposition REF ). This fact applies to \(D(m))\)  (Corollary REF ).
